Question: Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{-q^2 - 15q - 50}{-7q^2 - 77q - 210}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {-1(q^2 + 15q + 50)} {-7(q^2 + 11q + 30)} $ $ p = \dfrac{1}{7} \cdot \dfrac{q^2 + 15q + 50}{q^2 + 11q + 30} $ Next factor the numerator and denominator. $ p = \dfrac{1}{7} \cdot \dfrac{(q + 5)(q + 10)}{(q + 5)(q + 6)}$ Assuming $q \neq -5$ , we can cancel the $q + 5$ $ p = \dfrac{1}{7} \cdot \dfrac{q + 10}{q + 6}$ Therefore: $ p = \dfrac{ q + 10 }{ 7(q + 6)}$, $q \neq -5$